Understanding the difference between pre-image and inverse

The biggest difference between a preimage and the inverse function is that the preimage is a subset of the domain. The inverse (if it exists) is a function between two sets.

In that sense they are two very different animals. A set and a function are completely different objects.

So for example: The inverse of a function $f$ might be: The function $g:\mathbb R \to \mathbb R: g(x) = \sqrt[3]{x-9}$. Whereas the preimage of a set $B$ of the function might be $[1,3.5)\cup \{e, \pi^2\}$.

Now $g(x) = \sqrt[3]{x-9}$ and $[1,3.5)\cup \{7, \pi^2\}$ are completely different types of things.

This will be the case if $f$ is $f:\mathbb R \to \mathbb R: f(x) = x^3 + 9$ and $B= [10, 51.875) \cup \{352, \pi^6 + 9\}$.

The inverse $f^{-1}(x)$ (if it exist) is the function $g$ so that if $f(x) = y$ if and only if $g(y) = x$. So if $f(x) = x^3 + 9 = y$ then if such a function exists it must be that $g(y)^3 + 9 = y$ so $g(y)^3 = y-9$ and $g(y) = \sqrt[3]{y-9}$ so $g(x) = \sqrt[3]{x-9}$.

That's that.

The pre-image of $A= [10, 51.875) \cup \{352, \pi^6 + 9\}$ is the set $\{x\in \mathbb R| f(x) \in [10, 51.875) \cup \{352, \pi^6 + 9\}\}=$

$\{x\in \mathbb R| x^3 + 9 \in [10, 51.875) \cup \{352, \pi^6 + 9\}\}=$

$\{x\in \mathbb R| x^3 \in [1, 42.875) \cup \{343, \pi^6 \}\}=$

$\{x\in \mathbb R| x \in [1, 3.5) \cup \{7, \pi^2 \}\}=$

$[1, 3.5) \cup \{7, \pi^2 \}\}$.

And that's the other.

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Now that's not to say the inverse of a function and the pre-image of a set under the function aren't related. They are. But they refer to different concepts. This is similar to how a rectangle and its area are related. But one is a geometric shape... the other is a positive real number. THey are two different types of animals.

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I'll add more in an hour or so but I have to take the dog for a walk. I'll be back.

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It occurred to me as I was walking the dog that maybe what is confusing you is that the inverse function (if it exists) and the preimage of a set have very similar notation and the only way to tell them apart is in context.

If $f$ is invertible then the inverse function is written as $f^{-1}$ so if $f(x) = x^3 + 9$ then $f^{-1}(x) = \sqrt[3]{x-9}$.

But the preimage of $B$ under $f$ whether $f$ is invertible or or not is writen as $f^{-1}(B)$.

So if $f(x) = x^3 + 9$ then $f^{-1}(17) = 2$ means that if you enter $17$ into the function $\sqrt{x -9}$ you get $2$. But $f^{-1}(\{17\})=\{3\}$ and $f^{-1}(\{36,17\}) = \{2,3\}$ means that set of values that will output $\{17\}$ is the set $\{2\}$ and the set of values that will output $\{36,17\}$ is the set $\{2,3\}$.

A few things to note:

If $f$ is invertible then the preimage of a set is the same thing as the image of the set under the inverse function and that means the notation is compatible.

If $f(x) = x^3 + 9$ then $f^{-1}([1,36)) = [1,3)$ can be interpretated as both the the image of the set under the inverse function: $f^{-1}([1,36))= \{f^{-1}(x) = g(x) = \sqrt[3]{x-9}| x\in [1,36)\}$

OR it can be interpreted as the preimage for $f$: $f^{-1}([1,36)) = \{x\in \mathbb R| f(x) \in [1,36)\}$.

but this is not the case if $f$ is not invertible.

Say $f:\mathbb R \to [-1,1]; f(x)\to \sin x$. This is not invertible.

The pre-image of$B= \{\frac {\sqrt 2}2\}$ is $\{...-\frac {11\pi}4, -\frac {9\pi}4,-\frac{3\pi}4,-\frac \pi 4, \frac \pi 4, \frac {3\pi}4, \frac {9\pi}4, \frac {11\pi}4,....\}$ this is still written as $f^{-1}( \{\frac {\sqrt 2}2\})$ even though there is no function $f^{-1}:[-1,1]\to \mathbb R$.

Another thing to note is that not all the elements in $B$ have to have pre-image values.

If $f= x^2+9$ then $f^{-1}(\{8\}) = \emptyset$. This is because $\{x\in \mathbb R| f(x) = x^2 + 9 \in \{8\}\} = \emptyset$.

And some elements may have many preimages.

And $\sin^{-1}(\{\frac {\sqrt2} 2}$ showed.

When we talk about pre-image then it has two components: a set and a function. So when we say $f^{-1}[B]$, then we want the pre-image of the set $B$ (a subset of the co-domain) under the function $f$. So asking about pre-image of a function is a bit ambiguous.

Let us consider $f:\{1,2,3\} \rightarrow \{a,b,c\}$ such that $f(1)=a, f(2)=a$ and $f(3)=c$. Then \begin{align*} f^{-1}\left[\{a\}\right] & =\{1,2\}\\ f^{-1}\left[\{c\}\right] & =\{3\}\\ f^{-1}\left[\{a,c\}\right] & =\{1,2,3\}\\ f^{-1}\left[\{b\}\right] & =\emptyset\\ f^{-1}\left[\{a,b,c\}\right] & =\{1,2,3\} \end{align*}

The function $f$ (as described above) is not invertible. So relating the inverse function (which doesn't exist) to the inverse image is meaningless in this case.

One can check invertibility of a function $f: A \rightarrow B$ by checking the inverse images of singleton subsets of the co-domain.

What it means is that: if we can ensure that for every $b \in B$, the inverse image set $f^{-1}\left[\{b\}\right]$ has exactly one element (this is to ensure both one-one and ontoness), then $f$ is invertible.

So, there's no such thing as the preimage of a function. Functions can have inverses; functions do not have preimages.

An inverse is something that certain functions have, and the inverse of a function is another function.

Given a function, a preimage is something that sets have, and the preimage of a set is another set.

Specifically:

Given a function $f : A \to B$, that function may or may not have an inverse. If it does, then that inverse is a function $B \to A$.

Given a function $f : A \to B$, and a set $s$ which is a subset of $B$, that set always has a preimage under $f$. That preimage is a subset of $A$.

That might clear up some of your confusion.